Find $g'(x)$.

Show that $k = 6$.

Find the equation of the tangent to the graph of $y = g(x)$ at $x = 2$. Give your answer in the form $y = mx + c$.

Use your answer to part (a) and the value of $k$, to find the $x$-coordinates of the stationary points of the graph of $y = g(x)$.

Find $g’( - 1)$.

Hence justify that $g$ is decreasing at $x =  - 1$.

Find the $y$-coordinate of the local minimum.

$3{x^2} + 2kx - 15$     (A1)(A1)(A1)

Note:     Award (A1) for $3{x^2}$, (A1) for $2kx$ and (A1) for $- 15$. Award at most (A1)(A1)(A0) if additional terms are seen.

[3 marks]

$21 = 3{(2)^2} + 2k(2) - 15$     (M1)(M1)

Note:     Award (M1) for equating their derivative to 21. Award (M1) for substituting 2 into their derivative. The second (M1) should only be awarded if correct working leads to the final answer of $k = 6$.

Substituting in the known value, $k = 6$, invalidates the process; award (M0)(M0).

$k = 6$     (AG)

[2 marks]

$g(2) = {(2)^3} + (6){(2)^2} - 15(2) + 5{\text{ }}( = 7)$     (M1)

Note:     Award (M1) for substituting 2 into $g$.

$7 = 21(2) + c$     (M1)

Note:     Award (M1) for correct substitution of 21, 2 and their 7 into gradient intercept form.

OR

$y - 7 = 21(x - 2)$     (M1)

Note:     Award (M1) for correct substitution of 21, 2 and their 7 into gradient point form.

$y = 21x - 35$     (A1)     (G2)

[3 marks]

$3{x^2} + 12x - 15 = 0$ (or equivalent)     (M1)

Note:     Award (M1) for equating their part (a) (with $k = 6$ substituted) to zero.

$x =  - 5,{\text{ }}x = 1$     (A1)(ft)(A1)(ft)

Note:     Follow through from part (a).

[3 marks]

$3{( - 1)^2} + 12( - 1) - 15$     (M1)

Note:     Award (M1) for substituting $- 1$ into their derivative, with $k = 6$ substituted. Follow through from part (a).

$=  - 24$     (A1)(ft)     (G2)

[2 marks]

$g’( - 1) < 0$ (therefore $g$ is decreasing when $x =  - 1$)     (R1)

[1 marks]

$g(1) = {(1)^3} + (6){(1)^2} - 15(1) + 5$     (M1)

Note:     Award (M1) for correctly substituting 6 and their 1 into $g$.

$=  - 3$     (A1)(ft)     (G2)

Note:     Award, at most, (M1)(A0) or (G1) if answer is given as a coordinate pair. Follow through from part (c).

[2 marks]